First thanks for this page and the effort. The basic idea and the initial formulas are ok but after that the section "solve for Q1" and the factoring out and dividing by the coefficient are full of errors. Revise your math.

Quote: therefore we want to approximate cubic Bezier curves using quadratic Bezier curves, that will exactly match the original cubic curve for a specific value of (the vehicle's present position on the track curve). For algorithmic reasons, we want to replace each cubic curve with two quadratic curves.

Simpler (and probably less CPU expensive) solution:

1. divide the cubic at the chosen t param using de Casteljau algo (http://www.caffeineowl.com/graphics/2d/vectorial/bezierintro.html#cubicDivDeCastel)
2. approximate the two cubic resulted segments by using the mid-point-approximation. With (p0, p1, p2, p3) defining a cubic, the position of the Q1 control for the approx quad is (3·p2 - p3 + 3·p1 - p0)/4 (the quad will be - visually and in concerning the "trajectory speed" - no further apart than (sqrt(3)/36)·|p3 - 3·p2 + 3·p1 - P0| from the original cubic); apply this for the first and second cubic seg. See http://www.caffeineowl.com/graphics/2d/vectorial/cubic2quad01.html

--202.164.202.16 21:31, October 27, 2009 (UTC) Adrian

I barely remember writing this article nine years ago. I'm not surprised if I made some fundamental mathematical blunders. Thanks for the excellent feedback. I relinquish my interest in this article. Oktal (talk) 02:21, January 17, 2016 (UTC)